A Holistic Study of the Eclipsing Binary EQ Tau

JPAR Journal of Physics and Astronomy Research Vol. 1(3), pp. 015-034, November, 2014. © www.premierpublishers.org, ISSN: XXXX-XXXX

Research Article

A Holistic study of the eclipsing binary EQ Tau

M. M. Elkhateeb*,1,2 and M. I. Nouh1,2 E-mail: [email protected]

1Department of Astronomy, National Research Institute of Astronomy and Geophysics, 11421 Helwan, Cairo, Egypt 2Department of Physics, College of Science, Northern Border University, 1321 Arar, Saudi Arabia

We present a new BVR light curves of the system EQ Tau carried out in the period from March to April 2006 using a 50-cm F/8.4 Ritchey–Chretien telescope (Ba50) of the Baja Astronomical Observatory (Hungary), and 512 × 512 Apogee AP-7 CCD camera. The observed light curves were analyzed using the 2004 version of the Wilson-Devinney code. The results show that the more massive component is hotter than the low massive one by 100 K. A long term orbital period study shows that the period increases by the rate 8.946 (±0.365) x10-11 days/cycle. Evolutionary state of the system has been investigated and showed that, the primary component of the system is located nearly on the ZAMS for both the M-L and M-R relations. The secondary component is close to the TAMS track for M-L and above the M-R relations.

Key Words: Eclipsing binaries, contact, period variation, orbital solution, evolutionary status

INTRODUCTION

The variability of the system EQ Tau (G2, V=12.04, P=0.341349) was discovered by Tsesevich (1954). The system was included to the AAVSO list of eclipsing, while the first period was calculated by Whitney (1972), while the first CCD light curve was carried out in R band by Benbow and Mutel (1995). The system was monitored by Buckner, Nellermoe and Mutel (1998), and Nelson (2001). The first reliable spectroscopic elements of the system were estimated by Rucinski et al. (2001). Successive observations were carried out from 2000 to 2002 in BV band pass by Pribulla and Vanko (2002), and Vanko et al. (2004). They combined the calculated photometric elements resulting from their observations with published spectroscopic elements to yield the absolute parameters of the system. Light curves asymmetry was noted by Yang and Liu (2002) through their BV observations and they adopted the first spot model for the system EQ Tau. Many photometric studies were applied for the system light curve by Hrivnak et al. (2006), Yuan and Qian (2007) and Alton (2006, 2009). Recently, during the work in the present analysis, Li et al. (2014) presents a new light curve analysis for the system. In the present paper, we use the Wilson-Devinney (WD) code to estimate the physical parameters of the contact binary EQ Tau and to study its evolutionary status. A long term orbital period study was performed using O-C method and a possible connection between the light curve variation and the period change of the system was studied

OBSERVATIONS

Observations of EQ Tau were carried out on five nights from March to April 2006 in B, V and R band pass using a 50- cm F/8.4 Ritchey–Chretien telescope (Ba50) of the Baja Astronomical Observatory (Hungary), and 512 × 512 Apogee AP-7 CCD camera. Table (1) lists the coordinates of the variable, the comparison and the check stars. It’s clear that the comparison and check stars are close to the variable and they were in the same field, thus the extinction corrections can

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be ignored. The observed images were analyzed by the photometry software AIP4WIN (Berry and Buruell 2000) which is based on aperture photometry, including bias and dark subtraction and flat field correction. A complete light curves were obtained only in V and R filters. A total of 649 individual observations were obtained in VR band pass (252 in V and 397 in R), which covered a complete light curves in V and R bands, as displayed in Figure (1). The individual observations in both bands are listed in Table 2, with the magnitude difference (the variable minus the comparison). The phases were calculated using the ephemeris of Kreiner et al. (2000): Min I = 2440203.4343 + 0.d341347947 * E. (1) From the present observations, the times of three primary light minima were calculated using the Minima V2.3 Package (Nelson 2006) based on the Kwee and Van Worden (1956) fitting method. The calculated minima are shown in Table (3) and together with all published photoelectric and CCD minima in Table (4).

Table 1. Coordinates of EQ Tau, comparison, and the check stars

Star Name (2000.0) δ (2000.0) V B-V EQ Tau 03h48' 16.0'' +22o 17' 30.0'' 11.1 0.86 8 Comparison (TYC 1260-575-1) 03h48' 16.5'' +22o 17' 29.7'' 10.3 1.15 2 Check (GSC 01260-00800) 03h48' 17.4'' +22o 22' 43.0'' 8.19 1.05 0

Table 2. CCD Observations of EQ Tau in VR band.

V-band observations of EQ R-band observations of EQ Tau Tau JD Phase ΔR Error JD Phase ΔV Error 2453796. 0.08 0.9 0.03 2453796.3 0.076 1.22 0.05 3086 21 62 93 077 5 9 02 2453796. 0.08 0.9 0.03 2453796.3 0.081 1.19 0.04 3104 72 44 85 095 7 9 90 2453796. 0.09 0.9 0.03 2453796.3 0.086 1.17 0.04 3122 25 23 77 113 8 7 81 2453796. 0.09 0.8 0.03 2453796.3 0.092 1.16 0.04 3139 76 97 67 130 0 6 76 2453796. 0.10 0.8 0.03 2453796.3 0.097 1.14 0.04 3157 28 89 64 148 2 7 68 2453796. 0.10 0.8 0.03 2453796.3 0.102 1.13 0.04 3175 80 73 56 166 4 1 62 2453796. 0.11 0.8 0.03 2453796.3 0.107 1.11 0.04 3192 32 67 54 183 6 6 56 2453796. 0.11 0.8 0.03 2453796.3 0.112 1.10 0.04 3210 84 44 45 201 8 3 50 2453796. 0.12 0.8 0.03 2453796.3 0.118 1.08 0.04 3228 35 32 40 219 0 9 45 2453796. 0.12 0.8 0.03 2453796.3 0.123 1.08 0.04 3245 87 18 34 237 1 0 41 2453796. 0.13 0.8 0.03 2453796.3 0.128 1.07 0.04 3263 39 08 30 254 3 4 39 2453796. 0.13 0.7 0.03 2453796.3 0.134 1.05 0.04 3284 99 95 25 275 3 8 32 2453796. 0.14 0.7 0.03 2453796.3 0.139 1.04 0.04 3301 51 88 22 292 5 1 25 2453796. 0.15 0.7 0.03 2453796.3 0.144 1.04 0.04 3319 03 76 17 310 7 2 25

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Table 2. Cont. 2453796. 0.15 0.7 0.03 2453796.3 0.149 1.03 0.04 3337 55 65 12 328 9 2 21 2453796. 0.16 0.7 0.03 2453796.3 0.155 1.01 0.04 3354 07 57 09 346 1 9 16 2453796. 0.16 0.7 0.03 2453796.3 0.165 1 0.04 3372 58 51 07 381 4 000. 08 2453796. 0.17 0.7 0.03 2453796.3 0.170 0.99 0.04 3390 10 36 01 399 6 3 05 2453796. 0.17 0.7 0.03 2453796.3 0.175 0.98 0.04 3407 62 37 01 416 7 4 02 2453796. 0.18 0.7 0.02 2453796.3 0.180 0.98 0.04 3425 13 17 93 434 9 0 00 2453796. 0.18 0.7 0.02 2453796.3 0.201 0.95 0.03 3443 65 11 90 504 6 3 89 2453796. 0.19 0.6 0.02 2453796.3 0.206 0.95 0.03 3460 16 96 84 522 8 4 90 2453796. 0.19 0.6 0.02 2453796.3 0.217 0.95 0.03 3478 68 94 83 557 1 6 90 2453796. 0.20 0.6 0.02 2453796.3 0.227 0.95 0.03 3496 20 87 81 593 5 7 91 2453796. 0.20 0.7 0.02 2453796.3 0.232 0.96 0.03 3513 72 00 86 610 6 5 94 2453796. 0.21 0.6 0.02 2453796.3 0.237 0.96 0.03 3531 24 85 80 628 8 7 95 2453796. 0.22 0.6 0.02 2453796.3 0.243 0.96 0.03 3549 45 92 83 646 1 7 95 2453796. 0.22 0.6 0.02 2453796.3 0.248 0.96 0.03 3566 98 87 81 663 2 2 93 2453796. 0.23 0.6 0.02 2453796.3 0.253 0.96 0.03 3584 49 93 83 681 3 5 94 2453796. 0.24 0.6 0.02 2453796.3 0.258 0.96 0.03 3602 01 92 83 699 6 5 94 2453796. 0.24 0.6 0.02 2453796.3 0.268 0.97 0.03 3619 53 96 84 734 9 3 97 2453796. 0.25 0.6 0.02 2453796.3 0.274 0.97 0.03 3637 04 86 80 752 0 6 99 2453796. 0.25 0.6 0.02 2453796.3 0.279 0.98 0.04 3655 56 93 83 769 3 0 00 2453796. 0.26 0.6 0.02 2453796.3 0.284 0.99 0.04 3672 08 97 85 787 4 1 05 2453796. 0.26 0.6 0.02 2453796.3 0.301 0.98 0.04 3690 59 99 85 845 3 4 02 2453796. 0.27 0.6 0.02 2453796.3 0.306 0.98 0.04 3708 11 92 83 863 5 9 04 2453796. 0.27 0.7 0.02 2453796.3 0.311 0.99 0.04 3725 63 00 86 880 7 4 06 2453796. 0.28 0.6 0.02 2453796.3 0.322 1.00 0.04 3743 15 99 85 916 1 6 11 2453796. 0.28 0.7 0.02 2453796.3 0.327 1.02 0.04 3760 66 00 86 933 2 0 16 2453796. 0.29 0.7 0.02 2453796.3 0.332 1.02 0.04 3778 18 11 90 950 3 8 20 2453796. 0.29 0.7 0.02 2453796.3 0.337 1.03 0.04 3796 70 11 90 968 5 5 23 A Holistic study of the eclipsing binary EQ Tau

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Table 2. Cont. 2453796. 0.30 0.7 0.02 2453796.3 0.342 1.04 0.04 3818 36 17 93 986 6 6 27 2453796. 0.30 0.7 0.02 2453796.4 0.347 1.04 0.04 3836 88 22 95 003 8 8 28 2453796. 0.31 0.7 0.03 2453796.4 0.353 1.05 0.04 3854 40 43 03 022 1 2 30 2453796. 0.31 0.7 0.03 2453796.4 0.358 1.06 0.04 3871 91 48 05 039 3 9 36 2453796. 0.32 0.7 0.03 2453796.4 0.363 1.07 0.04 3889 43 50 06 057 5 8 40 2453796. 0.32 0.7 0.03 2453796.4 0.368 1.09 0.04 3907 95 59 10 075 6 6 47 2453796. 0.33 0.7 0.03 2453796.4 0.373 1.10 0.04 3924 47 54 08 093 9 7 52 2453796. 0.33 0.7 0.03 2453796.4 0.379 1.12 0.04 3942 97 59 10 110 0 1 58 2453796. 0.34 0.7 0.03 2453803.3 0.58 1.22 0.05 3959 48 79 18 499 70 4 00 2453796. 0.35 0.7 0.03 2453803.3 0.589 1.21 0.04 3977 01 77 17 506 1 1 94 2453796. 0.35 0.7 0.03 2453803.3 0.591 1.21 0.04 3995 52 84 20 514 5 6 96 2453796. 0.36 0.7 0.03 2453803.3 0.598 1.17 0.04 4013 05 92 23 539 7 5 80 2453796. 0.36 0.8 0.03 2453803.3 0.600 1.17 0.04 4030 57 09 30 546 7 3 79 2453796. 0.37 0.8 0.03 2453803.3 0.602 1.18 0.04 4048 09 23 36 553 8 6 84 2453796. 0.37 0.8 0.03 2453803.3 0.607 1.17 0.04 4066 60 24 36 570 8 5 80 2453796. 0.38 0.8 0.03 2453803.3 0.609 1.14 0.04 4084 13 43 44 577 8 7 68 2453796. 0.38 0.8 0.03 2453803.3 0.612 1.12 0.04 4101 65 57 50 586 5 2 58 2453802. 0.68 0.7 0.03 2453803.3 0.627 1.06 0.04 3605 84 39 02 638 8 1 33 2453802. 0.69 0.7 0.02 2453803.3 0.629 1.11 0.04 3616 16 31 98 646 9 7 56 2453802. 0.69 0.7 0.02 2453803.3 0.658 1.00 0.04 3626 47 24 96 742 2 6 11 2453802. 0.69 0.7 0.02 2453803.3 0.660 1.05 0.04 3637 79 22 95 749 2 2 30 2453802. 0.71 0.7 0.02 2453803.3 0.665 1.04 0.04 3688 29 09 90 766 1 1 25 2453802. 0.71 0.7 0.02 2453803.3 0.669 1.01 0.04 3699 60 22 96 781 6 3 14 2453802. 0.71 0.7 0.02 2453803.3 0.682 0.99 0.04 3710 93 04 87 825 6 3 05 2453802. 0.72 0.7 0.02 2453803.3 0.687 0.99 0.04 3721 25 13 91 841 3 3 05 2453802. 0.72 0.7 0.02 2453803.3 0.715 0.97 0.03 3732 57 15 92 793 2 6 99 2453802. 0.72 0.7 0.02 2453803.3 0.717 0.95 0.03 3747 99 07 89 494 3 7 91

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Table 2. Cont. 2453802. 0.73 0.6 0.02 2453803.4 0.739 0.96 0.03 3760 40 96 84 019 3 2 93 2453802. 0.75 0.6 0.02 2453803.4 0.763 0.95 0.03 3830 44 99 85 310 8 5 90 2453802. 0.75 0.7 0.02 2453803.4 0.773 0.96 0.03 3841 76 03 87 713 8 5 94 2453802. 0.77 0.7 0.02 2453803.4 0.787 0.96 0.03 3896 37 04 87 183 5 8 95 2453802. 0.77 0.7 0.02 2453803.4 0.812 0.98 0.04 3907 69 11 90 702 8 7 03 2453802. 0.79 0.7 0.02 2453803.4 0.820 0.97 0.03 3971 57 17 93 529 2 7 99 2453802. 0.82 0.7 0.03 2453803.4 0.822 0.99 0.04 4065 32 60 10 302 3 1 05 2453802. 0.83 0.7 0.03 2453803.4 0.841 1.03 0.04 4098 28 59 10 736 2 1 21 2453802. 0.84 0.7 0.03 2453803.4 0.845 1.04 0.04 4123 03 72 15 380 2 1 25 2453802. 0.84 0.7 0.03 2453803.4 0.847 1.06 0.04 4152 87 86 21 838 3 7 36 2453802. 0.85 0.8 0.03 2453803.4 0.849 1.04 0.04 4186 86 06 29 539 3 1 25 2453802. 0.86 0.8 0.03 2453803.4 0.851 1.04 0.04 4204 38 15 33 240 4 4 26 2453802. 0.86 0.8 0.03 2453803.4 0.873 1.05 0.04 4215 70 16 33 647 1 7 32 2453802. 0.87 0.8 0.03 2453803.4 0.877 1.07 0.04 4248 67 29 38 491 7 9 41 2453802. 0.87 0.8 0.03 2453803.4 0.899 1.16 0.04 4259 99 51 47 565 4 2 74 2453802. 0.88 0.8 0.03 2453815.3 0.883 1.10 0.04 4269 31 45 45 457 7 9 53 2453802. 0.88 0.8 0.03 2453815.3 0.890 1.13 0.04 4285 77 64 53 759 3 1 62 2453803. 0.58 0.9 0.03 2453815.3 0.89 1.14 0.04 3497 63 47 87 619 70 9 69 2453803. 0.58 0.9 0.03 2453815.3 0.902 1.16 0.04 3504 85 38 83 863 3 3 75 2453803. 0.59 0.9 0.03 2453815.3 0.903 1.19 0.04 3512 08 55 90 642 6 0 86 2453803. 0.59 0.9 0.03 2453815.3 0.905 1.17 0.04 3537 80 22 76 764 0 8 81 2453803. 0.60 0.8 0.03 2453815.3 0.915 1.21 0.04 3544 00 97 66 368 6 7 97 2453803. 0.60 0.9 0.03 2453815.3 0.91 1.22 0.05 3551 21 07 70 868 70 7 01 2453803. 0.60 0.9 0.03 2453815.3 0.918 1.22 0.05 3558 41 10 72 692 3 9 02 2453803. 0.60 0.8 0.03 2453815.3 0.922 1.25 0.05 3574 91 87 62 670 3 2 11 2453803. 0.62 0.8 0.03 2453815.3 0.924 1.26 0.05 3622 30 60 51 571 9 7 17 2453803. 0.62 0.8 0.03 2453815.3 0.930 1.27 0.05 3629 51 50 47 373 3 5 21 2453803. 0.62 0.8 0.03 2453815.3 0.931 1.28 0.05 3636 71 40 43 873 6 9 26 A Holistic study of the eclipsing binary EQ Tau

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Table 2. Cont. 2453803. 0.62 0.8 0.03 2453815.3 0.934 1.31 0.05 3643 92 14 32 774 3 1 35 2453803. 0.64 0.8 0.03 2453815.3 0.935 1.31 0.05 3696 47 16 33 751 6 6 37 2453803. 0.64 0.7 0.03 2453815.3 0.947 1.39 0.05 3704 70 94 24 792 6 9 71 2453803. 0.64 0.7 0.03 2453815.3 0.963 1.51 0.06 3714 99 84 20 784 5 9 20 2453803. 0.65 0.8 0.03 2453815.3 0.966 1.53 0.06 3723 25 11 31 685 2 3 26 2453803. 0.65 0.7 0.03 2453815.3 0.967 1.51 0.06 3730 46 78 18 860 6 3 18 2453803. 0.65 0.7 0.03 2453815.3 0.968 1.53 0.06 3739 74 88 22 586 9 4 26 2453803. 0.65 0.7 0.03 2453815.3 0.970 1.54 0.06 3747 95 86 21 869 2 9 32 2453803. 0.67 0.7 0.03 2453815.3 0.978 1.56 0.06 3798 45 51 07 789 2 6 39 2453803. 0.67 0.7 0.03 2453815.3 0.979 1.58 0.06 3805 66 67 13 901 5 0 45 2453803. 0.82 0.7 0.02 2453815.3 0.984 1.59 0.06 4300 16 33 99 919 8 2 50 2453803. 0.87 0.8 0.03 2453815.3 0.991 1.61 0.06 4489 69 48 46 942 5 6 60 2453803. 0.88 0.8 0.03 2453815.3 0.992 1.60 0.06 4504 15 24 36 794 8 3 54 2453803. 0.88 0.7 0.03 2453815.3 0.994 1.62 0.06 4515 47 97 25 951 2 7 64 2453803. 0.89 0.9 0.03 2453815.3 0.995 1.62 0.06 4549 45 11 72 695 5 4 63 2453803. 0.89 0.9 0.03 2453815.3 0.996 1.62 0.06 4556 67 07 70 960 8 3 63 2453814. 0.91 0.9 0.03 2453815.3 0.998 1.62 0.06 3484 79 74 98 596 1 2 62 2453814. 0.92 1.0 0.04 2453815.3 0.999 1.62 0.06 3518 76 25 19 969 5 4 63 2453814. 0.93 1.0 0.04 2453815.3 0.000 1.61 0.06 3551 74 75 39 497 9 1 58 2453814. 0.94 1.1 0.04 2453815.4 0.019 1.59 0.06 3585 73 35 63 037 4 2 50 2453814. 0.95 1.1 0.04 2453815.4 0.020 1.57 0.06 3618 71 82 83 042 8 7 44 2453814. 0.96 1.2 0.05 2453815.4 0.022 1.56 0.06 3652 69 52 11 704 1 3 38 2453814. 0.97 1.3 0.05 2453815.4 0.028 1.54 0.06 3685 67 19 39 069 8 2 30 2453814. 0.98 1.3 0.05 2453815.4 0.030 1.52 0.06 3719 65 49 51 407 1 5 23 2453814. 0.99 1.3 0.05 2453815.4 0.031 1.53 0.06 3752 63 76 62 078 4 6 27 2453814. 0.00 1.3 0.05 2453815.4 0.032 1.52 0.06 3785 61 67 58 308 8 3 22 2453814. 0.01 1.3 0.05 2453815.4 0.034 1.52 0.06 3819 59 54 53 808 1 1 21 2453814. 0.02 1.3 0.05 2453815.4 0.035 1.47 0.06 3852 57 23 40 092 4 2 01

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Table 2. Cont. 2453814. 0.03 1.2 0.05 2453815.4 0.036 1.5 0.06 3886 55 78 22 709 8 00 12 2453814. 0.04 1.1 0.04 2453815.4 0.039 1.49 0.06 3919 53 96 88 610 4 0 08 2453814. 0.05 1.1 0.04 2453815.4 0.040 1.44 0.05 3953 51 25 59 110 8 2 89 2453814. 0.06 1.0 0.04 2453815.4 0.042 1.45 0.05 3986 49 78 40 511 1 8 95 2453814. 0.07 1.0 0.04 2453815.4 0.043 1.45 0.05 4020 47 02 09 119 4 8 95 2453814. 0.08 0.9 0.04 2453815.4 0.046 1.41 0.05 4053 45 87 03 128 1 9 79 2453814. 0.09 0.9 0.03 2453815.4 0.047 1.41 0.05 4087 43 47 87 313 4 6 78 2453814. 0.10 0.9 0.03 2453815.4 0.048 1.40 0.05 4120 41 13 73 813 8 5 74 2453814. 0.11 0.8 0.03 2453815.4 0.050 1.41 0.05 4154 39 67 54 142 1 2 77 2453814. 0.12 0.8 0.03 2453815.4 0.051 1.38 0.05 4187 37 43 44 714 5 1 64 2453814. 0.13 0.8 0.03 2453815.4 0.052 1.38 0.05 4221 36 30 39 151 8 7 66 2453814. 0.14 0.7 0.03 2453815.4 0.054 1.36 0.05 4254 34 77 17 615 1 3 56 2453814. 0.15 0.7 0.03 2453815.4 0.055 1.35 0.05 4288 32 81 19 160 4 7 54 2453835. 0.36 0.8 0.03 2453815.4 0.056 1.37 0.05 3175 80 12 32 516 7 3 61 2453835. 0.37 0.7 0.03 2453815.4 0.058 1.31 0.05 3195 40 97 25 169 1 5 37 2453835. 0.37 0.8 0.03 2453815.4 0.070 1.27 0.05 3215 97 18 34 210 1 3 20 2453835. 0.38 0.8 0.03 2453815.4 0.071 1.27 0.05 3234 54 12 32 521 4 7 21 2453835. 0.39 0.8 0.03 2453835.3 0.396 1.16 0.04 3254 12 61 52 307 7 0 74 2453835. 0.39 0.8 0.03 2453835.3 0.402 1.17 0.04 3274 69 84 61 732 5 7 81 2453835. 0.40 0.8 0.03 2453835.3 0.408 1.18 0.04 3293 26 61 52 346 2 4 83 2453835. 0.40 0.8 0.03 2453835.3 0.41 1.21 0.04 3312 83 89 63 636 40 3 95 2453835. 0.41 0.8 0.03 2453835.3 0.419 1.25 0.05 3332 41 95 65 385 7 3 12 2453835. 0.41 0.9 0.03 2453835.3 0.425 1.25 0.05 3352 98 26 78 540 4 3 12 2453835. 0.42 0.9 0.03 2453835.3 0.431 1.30 0.05 3371 55 75 98 542 2 2 32 2453835. 0.43 0.9 0.03 2453835.3 0.436 1.32 0.05 3391 12 78 99 444 9 5 41 2453835. 0.43 1.0 0.04 2453835.3 0.442 1.34 0.05 3410 70 38 24 446 7 2 48 2453835. 0.44 1.0 0.04 2453835.3 0.448 1.42 0.05 3430 28 82 42 448 4 3 81 2453835. 0.44 1.1 0.04 2453835.3 0.454 1.44 0.05 3450 85 08 52 350 1 5 90

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Table 2. Cont. 2453835. 0.45 1.1 0.04 2453835.3 0.459 1.48 0.06 3469 42 13 54 352 8 0 04 2453835. 0.46 1.1 0.04 2453835.3 0.465 1.49 0.06 3489 00 12 54 542 6 4 10 2453835. 0.46 1.2 0.04 2453835.3 0.471 1.52 0.06 3508 57 07 93 256 3 1 21 2453835. 0.47 1.2 0.05 2453835.3 0.477 1.57 0.06 3528 14 33 03 581 0 8 44 2453835. 0.47 1.2 0.04 2453835.3 0.482 1.58 0.06 3548 72 09 94 160 8 1 45 2453835. 0.48 1.2 0.05 2453835.3 0.499 1.58 0.06 3567 29 92 28 659 9 3 46 2453835. 0.50 1.3 0.05 2453835.3 0.505 1.57 0.06 3626 01 17 38 967 7 3 42 2453835. 0.51 1.3 0.05 2453835.3 0.511 1.57 0.06 3665 15 00 31 969 4 0 41

0.6 V

0.8 R

)

- 1.2

(

1.4

1.6

1.8 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Phase Figure 1. Light curves of EQ Tau in V and R filters.

Table 3. Light minima of EQ Tau.

HJD Error Min Filter 2453814.3773 ±0.0002 I V 2453814.3774 ±0.0005 I B 2453815.3957 ±0.0003 I R 2453835.3610 ±0.0014 II V 2453835.3628 ±0.0004 II R

Orbital Period Behavior

Qian and Ma (2001) showed that the period of the system EQ Tau decreases by the rate -1.72 x 10-7days/year. Yang and Liu (2002), showed that the orbital period of the system exhibit a wavelike variation with cycle variation of 23 yr they refer the periodic change to a strong magnetic field or the presence of a third body orbits the system EQ Tau. Pribulla and Vanko (2002) interpreted the long term period changes as caused by the presence of a third body on a 50 yr orbit. A Holistic study of the eclipsing binary EQ Tau

J. Physics Astron. Res. 022

Alton (2006) estimated a very slow orbital period increase with a rate of 0.021 sec/year. Hrivnak et al. (2006) studied the period variation using the list of minima collected by Pribulla and Vanko (2002). They noted that the periodicity adopted by Pribulla and Vanko (2002) was based on older visual data of Tsesevich (1954) and photographic minima of Whitney (1972) from 1959 to 1962, while the period behavior based on photoelectric and recent photographic data showed a constant trend. They re-analyzed the period stability using high quality timings of minima of photoelectric observations together with their observed minima covering the interval from 1989 to 2005. Their results showed that the period of the system EQ Tau was constant around 1974 instead of cyclical variation. A cyclical behavior for the orbital period of the system EQ Tau was announced again by Yuan and Qian (2007) with a periodic variation of period 48.5 yr based on all published minima covering 64 yr. Alton (2009) showed that the period increased by 0.015 sec/yr over the interval from 2000-2009. From the previous orbital period studies of the system EQ Tau, it’s clear that the trend of period behavior depend on the quality of the collected minima and the covered interval. In the present paper we used an updated list of published minima from the literatures together with that listed in the web site (http:J//astro.sci.muni.cz/variables/ocgate/) and our new observed minima, which produced a complete set of data. A total of 264 timings of minima covered the interval of 59 years (~ 63130 revolutions) from 1954 to 2013 were used to follow the long term behavior of the system EQ Tau by means of (O-C) diagram. Linear ephemeris of Kreiner et al. (2000) (Eq. 1) was used to determine the calculated “C” values of the eclipse timings, listed in Table (4). Some collected uncertain minima were discarded in order to estimate accurate results. The (O-C) values were represented in Figure (2) versus the integer cycle E, where no distinctions have been made between primary and secondary minima. Scattering of some minima in the (O-C) diagram may be resulted from the variation in the observed light curves, which leads to non-asymmetry and also uncertainty in the calculated minima. Any mean light elements estimated by the linear best fit for all residual points are not suitable to estimate the times of minima in future. As the O-C diagram shows two ascending and similar descending branches, we are going to divide the (O-C) variation into four sections Ei – Ei-1, i = 1, 2, 3, 4. The period behavior in each section was represented by linear fit with corresponding change in the period ∆p, which were listed in Table (5) together with the standard deviations SD, correlation coefficient r and the residual sum of squares. Results in Table (5) reveal that the period of the system EQ Tau shows two intervals of increase and decrease, which looks like a periodic behavior. This behavior may be interpreted as caused by the presence of magnetic activity cycle and/or mass transfer mechanism. The general trend of the (O-C) diagram can be represented by a fifth degree polynomial with residual sum of squares = 0.0014 and correlation coefficient = 0.918 as:

Min I = 2440203.4314 + 0.341349197 * E – 4.4731*10-11 * E2 - 1.7186*10-15 * E3 + 6.4851*10-20 * E4 – 4.4607*10-25 * E5 (2)

Eq. 2 represents a new light element for the system EQ Tau which shows a period increases with rate dP/dE = 8.946 (±0.365) x 10-11 days/cycle or 9.566 (±0.391) x 10-8 days/year or 0.83 (±0.033) seconds/century. The (O-C)p residuals were calculating using polynomial ephemeris (Eq. 2) and listed in Table (4) and displayed in Figure(3).

Table 4. Times of minimum light for EQ Tau.

JD(Hel) E (O-C) (O- Mi Metho Ref JD(Hel) E (O-C) (O- Mi Metho Ref C)p n d . C)p n d . 2451849.18 3411 - - ccd 1 2455500.25 44813 - 0.001 ccd 35 74 7 0.014 0.001 p 06 0.009 2 P 8 2 3 2451849.18 3411 - - ccd 1 2455500.42 44813. - 0.002 ccd 35 75 7 0.014 0.001 p 22 5 0.008 1 S 7 1 3 2452219.54 3520 - - ccd 2 2455500.42 44813. - 0.002 ccd 35 85 2 0.016 0.002 p 23 5 0.008 2 S 2 0 2 2452219.54 3520 - - ccd 2 2455500.42 44813. - 0.002 ccd 35 86 2 0.016 0.001 p 23 5 0.008 2 S 1 9 2 2452219.54 3520 - - ccd 2 2455511.17 44845 - - ccd 35 90 2 0.015 0.001 p 19 0.011 0.000 S 7 5 1 7 A Holistic study of the eclipsing binary EQ Tau

Elkhateeb and Nouh 023

Table 4. Cont.

2452296.6934 35428 - - ccd 3 2455511.6870 44846.5 - 0.0015 ccd 14 p S 0.0160 0.0017 0.0080 2452614.6597 36359.5 - - ccd 3 2455520.7310 44873 - - ccd 15 S p 0.0153 0.0007 0.0097 0.0003 2452620.6330 36377 - - ccd 3 2455543.0906 44938.5 - 0.0019 ccd 35 p S 0.0156 0.0009 0.0084 2452684.6360 36564.5 - - ccd 3 2455539.6771 44928.5 - 0.0009 ccd 16 S S 0.0153 0.0006 0.0084 2453019.1580 37544.5 - 0.0006 ccd 1 2455551.1116 44962 - 0.0002 ccd 17 S p 0.0143 0.0091 2453351.9778 38519.5 - 0.0061 ccd 1 2455554.0125 44970.5 - - ccd 17 S S 0.0088 0.0097 0.0004 2453352.1414 38520 - - ccd 1 2455554.1835 44971 - - ccd 17 p p 0.0158 0.0010 0.0093 00010 2453352.1416 38520 - - ccd 1 2455570.5690 45019 - 0.0006 vis 18 p p 0.0156 0.0008 0.0085 2453358.1162 38537.5 - 0.0002 ccd 1 2455643.2755 45232 - - ccd 19 S p 0.0146 0.0091 0.0004 2453358.1164 38537.5 - 0.0004 ccd 1 2455643.2757 45232 - - ccd 19 S p 0.0144 0.0089 0.0002 2453814.3773 39874 0.0350 0.0495 ccd 4 2455643.2758 45232 - - ccd 19 p p 0.0088 0.0001 2453814.3774 39874 0.0351 0.0496 ccd 4 2455844.8368 45822.5 - - ccd 20 p S 0.0138 0.0062 2453815.3957 39877 0.0293 0.0438 ccd 4 2455846.8886 45828.5 - - ccd 21 p S 0.0101 0.0025 2453835.3610 39935.5 0.0258 0.0402 ccd 4 2455865.6584 45883.5 - - ccd 22 S S 0.0144 0.0069 2453835.3628 39936 - - ccd 4 2455885.6270 45942 - - vis 23 S p 0.1431 0.1286 0.0147 0.0073 2454057.3698 40586 - 0.0019 ccd 1 2455901.6750 45989 - - ccd 24 p p 0.0123 0.0100 0.0027 2454389.5007 41559 - 0.0006 ccd 5 2455910.0387 46013.5 - - ccd 25 p S 0.0129 0.0094 0.0021 2454389.5007 41559 - 0.0006 ccd 5 2455910.2091 46014 - - ccd 25 p p 0.0129 0.0096 0.0024 2454389.5009 41559 - 0.0008 ccd 5 2455921.6398 46047.5 - - ccd 26 p S 0.0127 0.0141 0.0069 2454506.2428 41901 - 0.0014 ccd 5 2455937.5132 46094 - - ccd 26 p p 0.0118 0.0134 0.0063 2454506.2435 41901 - 0.0021 ccd 5 2455963.6229 46170.5 - - ccd 27 p S 0.0111 0.0168 0.0099 2454509.3146 41910 - 0.0011 ccd 5 2456182.7685 46812.5 - - ccd 28 p S 0.0122 0.0166 0.0111 2454509.3147 41910 - 0.0012 ccd 5 2456222.3719 46928.5 - - ccd 29 p S 0.0121 0.0095 0.0043 2454509.3147 41910 - 0.0012 ccd 5 2456243.6999 46991 - - ccd 30 p p 0.0121 0.0158 0.0107 2455116.7407 43689.5 - - ccd 6 2456251.3860 47013.5 - - ccd 31 S S 0.0147 0.0035 0.0100 0.0050 2455148.3213 43782 - 0.0028 ccd 35 2456273.0610 47077 - - ccd 35 p p 0.0088 0.0106 0.0035 2455175.4577 43861.5 - 0.0014 ccd 7 2456273.0612 47077 - - ccd 35 S p 0.0096 0.0104 0.0033 2455175.4584 43861.5 - 0.0021 ccd 7 2456276.1329 47086 - - ccd 35 S p 0.0089 0.0108 0.0037

A Holistic study of the eclipsing binary EQ Tau

J. Physics Astron. Res. 024

Table 4. Cont. 2455175.4594 43861.5 - 0.0031 ccd 7 2456276.1333 47086 - - ccd 35 S p 0.0079 0.0104 0.0033 2455175.4594 43861.5 - 0.0031 ccd 7 2456276.6458 47087.5 - - ccd 32 S S 0.0079 0.0099 0.0052 2455175.6270 43862 - 0.0001 ccd 7 2456290.6423 47128.5 - - ccd 33 p S 0.0110 0.0087 0.0040 2455175.6286 43862 - 0.0017 ccd 7 2456291.4953 47131 - - ccd 34 p p 0.0094 0.0091 0.0044 2455186.3807 43893.5 - 0.0013 ccd 8 2456301.0512 47159 - - ccd 35 S p 0.0097 0.0109 0.0040 2455192.6950 43912 - 0.0006 ccd 9 2456301.0514 47159 - - ccd 35 p p 0.0104 0.0107 0.0038 2455219.3202 43990 - 0.0005 ccd 8 2456301.0514 47159 - - ccd 35 p p 0.0103 0.0107 0.0038 2455259.5991 44108 - 0.0002 ccd 10 2456301.0516 47159 - - ccd 35 p p 0.0105 0.0105 0.0036 2455453.4850 44676 - - ccd 8 2456301.2223 47159.5 - - ccd 35 p S 0.0102 0.0004 0.0105 0.0035 2455460.3146 44696 - 0.0031 ccd 35 2456301.2223 47159.5 - - ccd 35 p S 0.0075 0.0105 0.0035 2455460.3147 44696 - 0.0032 ccd 35 2456301.2224 47159.5 - - ccd 35 p S 0.0074 0.0104 0.0034 2455480.4519 44755 - - ccd 11 2456301.2225 47159.5 - - ccd 35 p S 0.0098 0.0002 0.0103 0.0033 2455485.7443 44770.5 - 0.0013 ccd 12 2456549.5516 47887 - - ccd 36 s p 0.0083 0.0118 0.0091 2455498.8846 44809 - - ccd 13 2456556.5502 47907.5 - - ccd 36 p S 0.0099 0.0003 0.0109 0.0082 2455499.3979 44810.5 - 0.0010 ccd 8 2456592.3899 48012.5 - - ccd 37 S S 0.0086 0.0127 0.0103 2455500.2505 44813 - 0.0011 ccd 35 2456613.7242 48075 - - ccd 38 p p 0.0090 0.0127 0.0105 2455500.2505 44813 - 0.0011 ccd 35 2456619.8641 48093 - - ccd 39 p p 0.0094 0.0170 0.0149

References: 1-.Yuan and Qian (2007); 2- Lehky (2009); 3- Hrivnak et al. (2006); 4- Present work; 5- Smelcer (2007); 6- Menzies (2009); 7- Agerer (2011); 8- Parimucha (211); 9- Diethelm (2010); 10- Samolyk (2010); Moschner (2012); 12- Samolyk (2010); 13- Diethelm (2011); 14- Poklar (2010); 15- Nelson (2011); 16- Simmon (2010); 17- Itoh (2010); 18- Stephan (2011); 19- Smelcer (2011); 20- Samolyk (2011); 21- Diethelm (2012); 22- Menzies (2011); 23- Stephan (2011); 24- Nelson (2012); 25- Shiokawa (2011); 26- Samolyk (2011); 27- Sabo (2012); 28- Menzies (2012a); 29- Parimucha (2013); 30- Menzies (2012b); 31- Hubscer (2013); 32- Menzies (2012c); 33- Diethelm (2013); 34- Vincenzi (2012); 35- Li et al. (2014); 36- Magris (2013); 37- UMA (2013); 38-Nelson (2013); 39-Samolyk (2013).

Table 5. Period behavior for the system EQ Tau

Intervals (2400000+) E to E E to E E to E E to E Parameters 0 1 1 2 2 3 3 4 30647 - 34389 34389 – 43483 - 52185 - 43483 52185 56301 ΔE (day) 3742 9094 8702 4116 Period (day) 0.341345116 0.341348869 0.341346785 0.341348580 ΔP (day) -2.831 x 10-6 9.220 x 10-7 -1.162 x 10-6 6.327 x 10-7 ΔP/P -8.293 x 10-6 2.701 x 10-6 -3.404 x 10-6 1.854 x 10-6 ΔP/ ΔE (d/cycle) -7.566 x 10-10 1.014 x 10-10 -1.335 x 10-10 1.537 x 10-10 Epoch (2400000+) 40203.3664 40203.4278 40203.4582 40203.3933 SD 0.00165 0.00269 0.00265 0.00116 R 0.99790 0.88708 0.94931 0.89122 Residual sum of 0.000003 0.00015 0.00032 0.00026 square R_Squared 0.99580 0.78691 0.90119 0.79427 A Holistic study of the eclipsing binary EQ Tau

Elkhateeb and Nouh 025

0.03

0.02

0.01

) 0

( -0.01

-0.02

-0.03

-30000 -20000 -10000 0 10000 20000 30000 40000 50000 60000 Integer cycle (E)

Figure 2. Period behavior of EQ Tau

0.015 0.01

p 0.005

) C

- 0 O

( -0.005 -0.01 -0.015 -40000 -20000 0 20000 40000 60000 Integer Cycle (E) Figure 3. Calculated residuals from the polynomial ephemeris

Light Curve Variation

The historical survey of the published light curves for the system EQ Tau since its discovery showed asymmetry and distortions at maximum phases, which seem to be referred to surface inhomogeneities of the components. The primary maximum (Max I) is brighter than the secondary one (Max II), which known as O’Connell effect, and appears in many eclipsing binaries which may refer to hot or cool regions on either components. Observations by Yang and Liu (2002) showed a typical O’Connell effect and a primary maximum brighter than secondary one by 0.03 mag., while Pribulla and Vanko (2002) showed maximum difference of about 0.017. Observed light curves by Hrivnak et al (2006) displayed a flat-bottom in the secondary minimum which indicates that the cooler component was eclipsed totally and showed a maximum differences (O’Connell effect) of about ~ 0.03 mag. Alton (2006, 2009) observed an apparent asymmetry at Max I and linked it to hot or dark spot on either components facing the observer. A noticeable variation in the light curves observed by Yuan and Qian (2007) in different seasons, they explained this variation as the result of a strong and variable dark spot activity. Most of the previous observations together with our light curves indicate some distortions in the two maximum phase (Max I and Max II) and also magnitude difference between them (O’Connell effect). This phenomenon was observed in many contact binaries (i.e. AQ Tuc (Hilditch and King (1986), and CN And Keskin (1989)). They explained it as a pulsation of a common envelope due to mass transfer between two components (Li et al. 2002). The magnitude difference between the two maxima and apparent asymmetry may be arising from hot star spot(s) on either component (which increases the flux during Max I) Alton (2006), or dark star spot(s) which decrease the level of Max II A relation between the light curve variation and orbital period changes in the same cycle was predicted by Applegate (1992). A Holistic study of the eclipsing binary EQ Tau

J. Physics Astron. Res. 026

Using our observations together with all published light curves, the light levels (Max I, Max II, Min I, and Min II) were estimated. The amplitudes (depths) of the primary Ap (mag(Min I – Max I)) and secondary As (mag(Min II – Max I)), and the magnitude difference between both maxima (O’Connell effect) Dmax (mag(Max I – Max II)) and minima Dmin (mag(Min I – Min II)) have been calculated for each light curve and listed in Table (6), while displayed in Figure(4). The behavior of the parameters Dmax, Dmin, Ap, and As showed a wave-like variation as periodic function. This behavior may be interpreted by a periodic effect of some physical mechanism (i.e. magnetic activity, and/or mass transfer). It’s clear that both the period change behavior and light curve parameters (Dmax, Dmin, Ap, and As) of the system EQ Tau showed a periodic variation which may be referred to the presence of magnetic activity cycles in more massive components and/or mass transfer mechanism. A future sequence series of observations for the system EQ Tau are needed in order to follow this periodic variation.

Table 6. Light curve parameters for EQ Tau

D D A A Ref JD Date max min p s (mag) (mag) (mag) (mag) . 245184 2000.8 0.0040±0.000 0.0640±0.002 0.6860±0.028 0.6220±0.025 1 9 2 6 0 4 245194 2001.0 - 0.0840±0.003 0.7160±0.029 0.6320±0.025 1 1 8 0.0290±0.001 4 2 8 2 245207 2001.5 0.0170±0.000 0.0740±0.003 0.6780±0.027 0.6040±0.024 2 2 7 0 7 7 245224 2001.9 - 0.1100±0.004 0.7700±0.031 0.6600±0.026 3 0 0.0300±0.001 5 4 9 2 245263 2003 - 0.1000±0.004 0.7750±0.031 0.6750±0.027 4 9 0.0350±0.001 1 6 6 4 245335 2005 - 0.0690±0.002 0.7070±0.028 0.6380±0.026 1 5 0.0490±0.002 8 9 1 0 245377 2006.1 - 0.0350±0.001 0.4450±0.0 0.4100±0.0 5 0 0.0400±0.001 4 182 167 6 245382 2006.3 0.0030±0.000 0.0545±0.002 0.6855±0.028 0.6310±0.025 6 1 1 2 0 8 245451 2008.1 - 0.0280±0.001 0.4850±0.019 0.5130±0.020 7 4 0.0450±0.001 1 8 9 8 245550 2010.8 - 0.0016 0.0302 0.7000±0.028 8 0 0.0250±0.001 0.0400± 0.7400± 6 0 245630 2013.1 - 0.0700±0.002 0.0302 0.6700±0.027 8 1 0.0250±0.001 9 0.7400± 4 0 Reference:1-Yuanand Qian (2007), 2- PribullaandVanko (2002), 3- YangandLiu (2002), 4- Hrivnak et al. (2006), 5- Alton (2006), 6- This Paper, 7- Alton (2009), 8- Li et al. (2014).

A Holistic study of the eclipsing binary EQ Tau

Elkhateeb and Nouh 027

0.12 0.04

0.1 0.02

0.08

n i

a 0.06

m D m -0.02

D 0.04 -0.04 0.02 b -0.06 a 0 2002 2006 2010 2014 2000 2002 2004 2006 2008 2010 2012 2014 2000 2004 2008 2012 Date Date

0.9 0.8

0.8 0.7

0.7

0.6

p s

A 0.6 A

0.5 0.5

0.4 d c 0.4

0.3 2000 2002 2004 2006 2008 2010 2012 2014 2000 2002 2004 2006 2008 2010 2012 2014 Date Date

Figure 4 Variations of Dmax, Dmin, Ap, and As,for the system EQ Tau in V filter

Variation of the brightness difference between both maxima (Dmax) and minima (Dmin) (Fig. 6. a, b.) can be represented by a fitting as seen in Eq. 3 and 4 respectively as:

Dmax = -0.0210 + 0.0356 *COS (2.2417*t – 150.7340) (3) Dmin = 0.0525 + 0.2886*COS (0.5131*t – 2.9724) (4)

Behavior of the amplitude (depth) of the primary Ap and secondary As(Fig. 6. c, d) can be represented by a sinusoidal fit as seen in Eq. 5 and 6 respectively as:

Ap = 0.6230 + 0.1723 *COS (0.6870*t – 345.4671) (5) As = 0.5617 + 0.1249 *COS (0.6905*t – 352.2458) (6)

Light Curve Modeling

Although the system EQ Tau was discovered since 1954 (60 years ago) it didn’t take sufficient interest in observation and light curve modeling. The first photometric solution was published by Pribulla and Vanko (2002) based on photoelectric observations in BV band pass. Although the asymmetry was clear in the observed light curves, they ruled out any spotted solution due to the low quality of their observations. A series of photometric solution were announced by Yang and Liu (2002), Vanko et al. (2004), Hrivnak et al. (2006), Alton (2006 and 2009), Yuan and Qian (2007) and Li et al. (2014). Because of the clear asymmetry in all published light curves of the system EQ Tau, the previous photometric solutions are in agreement with the presence of spot(s) on either or both system components. The only difference was the type of the spot(s) (hot or cool) and their distribution on the star’s surface. In this paper we present a photometric solution using the 2004 version of the Wilson-Devinney (WD) light curve and radial velocity analysis program with the Kurucz atmospheres (Wilson and Devinney, 1971, Wilson, 1990, Kallrath, et al., 1998) as implemented in the Windows software WDwint (Nelson, 2009) to analyze the data. We analyzed the individual light curve observations instead of normal light curves, which don’t reveal a real light variation of the system. The surface temperature of the primary star was fixed at 5800 K and the logarithm law of Van Hamme (1993) table was used to adopt and interpolate the bolometric limb darkening coefficients (x1 = x2, y1 = y2) and model atmosphere was applied. We adopted g1 = g2 = 0.32 (Lucy 1967) and the albedo value A1 = A2 = 0.5 (Rucinski 1969). During the execution of WDint56a Package, Mode 3 (overcontact) was applied and some parameters were held fixed (i.e. T1 = 5800 K A Holistic study of the eclipsing binary EQ Tau

J. Physics Astron. Res. 028

(according to the system spectral type G2 (Popper 1980), g, A, x1 = x2). The adjusted parameters are the inclination i and temperature T2 of the secondary component, the surface potentials Ω1 = Ω2, the mass ration q (M2/M1), and the monochromatic luminosity L1 of star 1 (the relative luminosity of star 2 was calculated by the stellar atmosphere model). The accepted model reveals modified parameters which are listed in Table (7) with their formal errors estimated by the fit and represented in Figure (5). It’s clear that the accepted model included three hot spots (two on the primary component and only one on the secondary, also the more massive component is hotter than the low massive one by ∆T ~ 100 K. Based on the calculated parameters a three dimension geometric structure of the system EQ Tau was constructed using the software package Binary Maker 3.03 (Bradstreet and Steelman 2002) (Figure (6)). The absolute physical dimensions were calculated and are listed in Table (8) together with all published parameters calculated by previous photometric solutions.

Table 7. Photometric solution EQ Tau.

Parameter Element i(0) 82.04±0.85 g1 = g2 0.5 A1 = A2 0.32 q (M2 / M1) 0.4453±0.0014 Ω1= Ω2 2.7399±0.0047 Ωin 2.7691 Ωout 2.5001 T1 (K) 5800 Fixed T2 (K) 5701±5 r1 pole 0.4291±0.0002 r1 side 0.4580±0.0003 r1 back 0.4874±0.0005 r2 pole 0.2962±0.0006 r2 side 0.3097±0.0007 r2 back 0.3460±0.0013

Spot parameters for star 1

Spot A: Co-latitude (deg) 135.3±5.524 Longitude (deg) 146.7±5.989 Spot radius (deg) 9.100±0.372 Temp. factor 1.100±0.045

Spot B: Co-latitude (deg) 122±4.981 Longitude (deg) 75±3.062 Spot radius (deg) 11.24±0.459 Temp. factor 1.35±0.055

Spot parameters for star 2

Co-latitude (deg) 118±4.817 Longitude (deg) 97.7±3.989 Spot radius (deg) 15.3±0.625 Temp. factor 1.43±0.058

∑ (O-C)2 0.04936

A Holistic study of the eclipsing binary EQ Tau

Elkhateeb and Nouh 029

V 0.9

0.8

x 0.7

e z

i 0.6

r o

N 0.5

0.4

0.3

0.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Phase Figure 5.Observed light curves (filled circles), and the synthetic curves (solid line)

Table 8. Physical parameters of EQ Tau.

Parameter Ref M1(M⊙) M2 (M⊙) R1(R⊙) R2(R⊙) L1 (L⊙) L2 (L⊙) log T1 log T2 . 1.320±0. 0.590±0. 1.160±0. 0.820±0. 1.350±0. 0.640±0. 3.763±0. 3.758±0. 1 054 024 047 034 055 026 154 153 1.217±0. 0.537±0. 1.139±0. 0.786±0. 1.370±0. 0.649±0. 3.768±0. 3.767±0. 2 05 022 047 032 056 027 154 154 1.280±0. 0.570±0. 1.170±0. 0.810±0. 1.390±0. 0.630±0. 3.763±0. 3.758±0. 3 052 023 048 033 057 026 154 153 1.230±0. 0.540±0. 1.140±0. 0.790±0. 1.330±0. 0.610±0. 3.763±0. 3.760±0. 4 05 022 047 032 054 026 154 154 1.214±0. 0.541±0. 1.136±0. 0.787±0. 1.310±0. 0.600±0. 3.763±0. 3.756±0. 5 05 022 046 032 054 025 154 153 Reference: 1- Yang and Liu (2002), 2- Vanko et al (2004), 3- Hrivnak et al. (2006), 4- Yuang and Qian (2007), 5- This Paper

Phase 0.68 Phase 0.2

Figure 6. Geometric structure of the binary system EQ Tau.

A Holistic study of the eclipsing binary EQ Tau

J. Physics Astron. Res. 030

DISCUSSION AND CONCLUSION

New BVR CCD observations were carried out for the system EQ Tau which added a new five minima. We have used all published times of minima from 1954 to 2013 (~ 59 yr) ~ 63130 revolution, together with our new minima to study the long term orbital period behavior of the system EQ Tau. The period shows increase with a rate of dP/dE = 8.946 (±0.365) x 10-11 days/cycle or 9.566 (±0.391) x 10-8 days/year or 0.83 (±0.033) seconds/century. Long term stability of the light curves of the system EQ Tau shows a periodic change in the magnitude difference between both maxima Dmax(t) (O’Connell effect) and minima Dmin(t), the amplitude (depths) of the primary Ap(t) and secondary As(t). A synchronous variation in both orbital period and light curve parameters Dmax(t), Dmin(t), Ap(t) and As(t) for the system EQ Tau may be interpreted by the presence of magnetic activity cycle and/or mass transfer mechanism. A photometric solution of the observed light curves by means of W-D code showed that the more massive component is hotter than the low massive one by ~ 100 K. We used the physical parameters listed in Table (8) to investigate the current evolutionary status of EQ Tau. In Figures (7) and (8), we plotted the components of EQ Tau on the mass–luminosity (M-L) and mass–radius (M-R) relations along with the evolutionary tracks computed by Girardi et al. (2000) for both zero age main sequence stars (ZAMS) and terminal age main sequence stars (TAMS) with metalicity z  0 . 0 1 9. Figure (9) displays the position of the components of EQ Tau on the effective temperature – luminosity diagram of Ekstrom et al. (2012). As it is clear from the figures, the primary component of the system is located nearly on the ZAMS for both the M-L and M-R relations. The secondary component is close to the TAMS track for M-L and above the M-R relations. To locate components on the Teff-luminosity relation for single stars, we used for this purpose, the non-rotated evolutionary models of Ekström et al. (2012) in the range 0.8–120 M⊙ at solar metalicity (z = 0.014). We used the tracks for the masses 0.9 M⊙, 1 M⊙ and 1.1 M⊙. The highly poor fit of the secondary and the fair fit of the primary reflect the contact nature of the system. The mass-effective temperature relation (M–Teff) relation for intermediate and low mass stars (Malkov, 2007) is displayed in Figure (10). The location of our mass and radius on the diagram revealed a good fit for the primary and poor fit for the secondary components. This gave the same behavior of the system on the mass-luminosity and mass-radius relations.

Primary 4 Secondary TAMS 3

2 ZAMS



L 1

o 0

-1

-2

-3

-4 -1 -0.5 0 0.5 1 log M/M  Figures 7. The position of the components of EQ Tau on the mass–luminosity diagram. The filled circle denotes the primary and the open circle represents the secondary.

A Holistic study of the eclipsing binary EQ Tau

Elkhateeb and Nouh 031

1.25 Primary

Secondary TAMS

0.75



/ ZAMS

R 0.25

-0.25

-0.75

-1 -0.5 0 0.5 1 log M/M  Figures 8. The position of the components of EQ Tau on the mass– radius diagram. The filled circle denotes the primary and the open circle represents the secondary.

2.5 Primary Secondary 2

) 1.5



(

L 1

o l 0.5 1.1 M

0 1 M 0.9 M -0.5

-1 3.5 3.6 3.7 3.8 3.9

log Teff Figure 9. The position of the components of EQ Tau on the effective temperature – luminosity diagram of Ekstrom et al. (2012) .

A Holistic study of the eclipsing binary EQ Tau

J. Physics Astron. Res. 032

1.2 Primary Secondary 0.8



T 0.4

-0.4 -1.5 -1 -0.5 0 0.5 1 1.5 2

log M / M Figure 10. Position of the components of EQ Tau on the empirical mass-Teff relation for low-intermediate mass stars by Malkov (2007).

ACKNOWLEDGEMENTS

This research has made use of the NASA’s ADS, AAVSO data base and the available on-line material of the IBVS. Our sincere thanks to Dr. Tibor Hegedus and Dr. Biro Barna from Baja Astronomical Observatory, Hungary for their kind help during the observations.

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Accepted 10 November, 2014.

Citation: Elkhateeb MM, Nouh MI (2014). A Holistic study of the eclipsing binary EQ Tau. Journal of Physics and Astronomy Research, 1(3): 015-034.

Copyright: © 2014 Elkhateeb and Nouh. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

A Holistic study of the eclipsing binary EQ Tau